The v-topology

Abstract

This chapter describes the v-topology. It develops a powerful technique for proving results about diamonds. There is a topology even finer than the pro-étale topology, the v-topology, which is reminiscent of the fpqc topology on schemes but which is more “topological” in nature. The class of v-covers is extremely general, which will reduce many proofs to very simple base cases. The chapter provides a sample application of this philosophy by establishing a general classification of p-divisible groups over integral perfectoid rings in terms of Breuil-Kisin-Fargues modules. Another use of the v-topology is to prove that certain pro-étale sheaves on Perf are diamonds without finding an explicit pro-étale cover.